Prove that the homogeneous coordinates of any point on a conic may be taken to be \((t^2, t, 1)\), where \(t\) is a parameter. The quadratic equations \(Q \equiv at^2+2bt+c=0\), \(Q' \equiv a't^2+2b't+c'=0\) determine two pairs of points on the conic; find
The equation of a conic referred to rectangular Cartesian coordinate axes is \[ ax^2+2hxy+by^2+2gx+2fy+c \equiv (a,b,c,f,g,h)(x,y,1)^2=0; \] prove that the line \(lx+my+1=0\) touches the conic, if \(\Sigma \equiv (A,B,C,F,G,H)(l,m,1)^2=0\), where \(A, B, \dots\) are the cofactors of \(a, b, \dots\) in the determinant \[ \delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \] which is assumed not to be zero. Prove also that, if the line \(lx+my+1=0\) is a directrix of the conic, \[ \frac{l^2-m^2}{a-b} = \frac{lm}{h} = \frac{\Sigma}{\delta}, \] and interpret these equations geometrically when \(a=b, h=0\).
The set of numbers \(x_1, x_2, \dots, x_n\) are transformed into the set of numbers \(\xi_1, \xi_2, \dots, \xi_n\) by means of the equations \begin{align*} \xi_1 &= a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n, \\ &\vdots \\ \xi_n &= a_{n1}x_1 + a_{n2}x_2 + \dots + a_{nn}x_n; \end{align*} the set of numbers \(y_1, y_2, \dots, y_n\) are transformed into the set of numbers \(\eta_1, \eta_2, \dots, \eta_n\) in the same way, and the set of numbers \(\pounds_1, \pounds_2, \dots, \pounds_n\) are similarly transformed into the set of numbers \(X_1, X_2, \dots, X_n\). Shew that, if all the coefficients \(a_{ij}\) are real, and if \(a_{ij}=a_{ji}\), then
Assume the theorem:
"If \(f'(x)\) exists for \(a \le x \le b\), then
\[ f(b)-f(a)=(b-a)f'(\xi), \]
where \(a<\xi
Define the area bounded by a closed curve. Obtain an expression for this area. A closed curve is given by the equation \[ xy=f\left(\frac{y}{x}\right). \] Shew that the area enclosed by the curve is given by the integral \(\frac{1}{2}\int f(\xi)\frac{d\xi}{\xi}\) taken between suitable limits. Find the area enclosed by one loop of the curve \[ x^4y^2+y^4+4xy+3x^2=0. \]
Prove that two couples of equal moment in the same or in parallel planes are equivalent to each other, and that a number of couples in parallel planes acting jointly on a rigid body are equivalent to a single couple of moment equal to the algebraic sum of the moments of the couples. Prove that a set of forces acting in one plane on a rigid body has the same effect upon it as a single force or a single couple. Emphasize the fundamental assumptions made during the proof. A uniform triangular lamina \(ABC\) of weight \(W\) can turn in a horizontal plane about a frictionless pivot at a point \(O\). Its weight is borne by three short pegs attached to it at the corners \(A, B, C\) which rest on a rough horizontal plane, the coefficient of friction between each of them and the plane being \(\mu\). Shew that the couple required to cause the lamina to turn about \(O\) is \(\frac{1}{3}\mu W(OA+OB+OC)\) and that there is no reaction at \(O\) if the sides of the lamina all subtend angles \(2\pi/3\) at \(O\).
Two heavy equal uniform rods, each of weight \(W\), stand in a vertical plane on a rough horizontal plane (coefficient of friction \(\mu\)), their upper ends being smoothly jointed together. The angle between the rods is \(2\alpha\). A weight \(2W\) is hung by two equal strings from the middle points of the two rods, the lengths of the strings being such that they each make an angle \(\beta\) with the vertical. Shew that for limiting equilibrium \[ \tan\beta = 2\tan\alpha \pm 4\mu, \] and that the feet of the rods will be drawn together if \(\tan\beta > 2\tan\alpha+4\mu\) or will spread apart if \(\tan\beta < 2\tan\alpha-4\mu\).
A particle is projected from a point on the ground so as to pass just over a vertical wall of height \(h\) at horizontal distance \(a\) from the point of projection. Shew that it strikes the ground at a distance \(2u^2h/ga\) beyond the wall, \(u\) being the horizontal component of velocity. Shew that in order that the particle may clear the wall the total velocity of projection must be at least equal to \(\sqrt{\{g(h+l)\}}\), where \(l^2=a^2+h^2\): and that if the total velocity of projection is \(\sqrt{\{g(h+k)\}}\), where \(k>l\), then \(u^2\) must lie between the values \[ \frac{ga^2}{2l^2}\{k \pm \sqrt{(k^2-l^2)}\}. \]
A bead of mass \(m\) slides on a fixed rough circular wire of radius \(a\), the coefficient of friction being \(\mu\). If the forces acting on the bead are in the plane of the circle, find a differential equation connecting the time with the position of the bead on the wire. Consider in particular the case in which the only forces on the bead are a constant attraction \(ma\omega^2\) towards the centre of the circle, and the reaction of the wire. The bead is projected with velocity \(a\Omega\). Shew (i) that if \(\Omega > \omega\) the bead ultimately moves with constant velocity \(a\omega\), and (ii) that if \(\Omega < \omega\) the bead comes to rest after the radius through it has turned through an angle \[ \frac{1}{2\mu}\log\frac{\omega^2}{\Omega^2}. \]
Two particles of masses \(m\) and \(m'\) are attached to the ends of a spring of natural length \(l\) and modulus \(\lambda\). They fall with the spring at its natural length and vertical so that \(m'\) strikes a horizontal inelastic table when both masses are moving vertically downwards with velocity \(V\). Shew that if \(m'\) leaves the table it does so when the velocity of \(m\) is \(V_1\), given by \[ mV_1^2 = mV^2 - m'(m'+2m)lg^2/\lambda, \] and that in the subsequent motion the extension of the spring varies harmonically in the period \(2\pi\sqrt{\{mm'l/\lambda(m+m')\}}\), the maximum extension being \[ \left(\frac{mm'l\{V^2-m'lg^2/\lambda\}}{\lambda(m+m')}\right)^{\frac{1}{2}}. \]